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Workbook for Stewart’s "Calculus"

Cory Wilson

Section 3.4 Limits at Infinity & Horizontal Asymptotes

Subsection 3.4.1 Before Class

https://mymedia.ou.edu/media/3.4-1/1_ptc8pjgb
Figure 25. Pre-Class Video 1

Subsubsection 3.4.1.1 The Ideas

Example 3.4.1.
Sketch the graph of \(f(x) = \dfrac{x^2-1}{x^2+1}\) using the techniques of Section 3.3.
The graph should look like this:
This is the graph of \(f(x) = \dfrac{x^2-1}{x^2+1}\)
Definition 3.4.2. Limits at Infinity.
Let \(f\) be a function defined on some interval \((a,\infty)\text{.}\) Then \(\lim_{x\to\infty} f(x) = L\) means that the values of \(f(x)\) can be made arbitrarily close to \(L\) by requiring \(x\) to be sufficiently large.
If \(g\) is defined on some interval \((-\infty,a)\text{,}\) then \(\lim_{x\to-\infty} g(x) = L\) means that the values of \(g(x)\) can be made arbitrarily close to \(L\) by requiring \(x\) to be sufficiently large negative.
We read the limits above (for \(f\)) as
  • the limit of \(f(x)\text{,}\) as \(x\) approaches \(\infty\text{,}\) is \(L\)
  • the limit of \(f(x)\) as \(x\) increases without bound, is \(L\)
with the obvious changes for \(g\)
Definition 3.4.3. Horizontal Asymptote.
The line \(y=L\) is called a horizontal asymptote of the curve \(y=f(x)\) if either
\begin{equation*} \lim_{x\to\infty} f(x) = L\qquad \text{or}\qquad \lim_{x\to -\infty} f(x) = L \end{equation*}
Example 3.4.4.
Write the horizontal asymptotes of the function \(f(x) = \dfrac{x^2-1}{x^2 + 1}\)
Solution.
\(y=1\)

Subsection 3.4.2 Pre-Class Activities

Example 3.4.5.

Write the horizontal asymptotes of the function \(f(x) = \dfrac{3x-2}{2x+1}\text{.}\)
Solution.
\(y = \dfrac{3}{2}\)

Example 3.4.6.

Does the function \(f(x) = \dfrac{1-x^2}{x^3-x+1}\) have any horizontal asymptotes? If it does, give their equation. If it doesn’t, explain why.
Solution.
Yes, \(y=0\)

Example 3.4.7.

The function \(f(x) = \dfrac{x-9}{\sqrt{4x^2 + 3}}\) has two horizontal asymptotes: \(L = \dfrac{1}{2}\) and \(L = - \dfrac{1}{2}\text{.}\) Use limit notation to describe the horizontal asymptotes.
Solution.
\begin{align*} \ds \lim_{x\to\infty} f(x) &= \dfrac{1}{2}\\ \ds \lim_{x\to -\infty} f(x) &= -\dfrac{1}{2} \end{align*}

Subsection 3.4.3 In Class

Subsubsection 3.4.3.1 Computing Limits at Infinity

Question 3.4.8.
Think about \(\ds \lim_{x\to\infty} \dfrac{1}{x}\) and \(\ds \lim_{x\to -\infty} \dfrac{1}{x}\text{.}\) What do you expect these limits to be? Why? What about \(\ds \lim_{x\to \pm \infty} x^r\text{,}\) for some \(r \gt 0\text{?}\)
Solution.
Answers vary
Theorem.
If \(r\gt 0\) is a rational number, then \(\ds \lim_{x\to \infty} \dfrac{1}{x^r} = 0\text{.}\) If \(r \gt 0\) is a rational number such that \(x^r\) is defined for all \(x\text{,}\) then \(\ds \lim_{x\to -\infty} \dfrac{1}{x^r} = 0\)
Example 3.4.9.
Evaluate \(\ds \lim_{x\to \infty} \dfrac{3x^2-x-2}{5x^2+4x+1}\)
Solution.
\(\ds \lim_{x\to \infty} \dfrac{3x^2-x-2}{5x^2+4x+1} = \dfrac{3}{5}\)
Example 3.4.10.
Find the asymptotes of \(f(x) = \dfrac{\sqrt{2x^2+1}}{3x-5}\)
Solution.
\(y = \pm \dfrac{\sqrt{2}}{3}\)
Example 3.4.11.
Compute \(\ds \lim_{x\to \infty} (\sqrt{x^2+2}-x)\)
Solution.
\(\ds \lim_{x\to \infty} (\sqrt{x^2+2}-x)=0\)
Example 3.4.12.
Find the following limits, or argue why it doesn’t exist:
  1. \(\displaystyle \ds \lim_{x\to -\infty} \dfrac{4x^3 + 6x^2 - 2}{2x^3 - 4x + 5}\)
  2. \(\displaystyle \ds \lim_{x\to -\infty} \dfrac{\sqrt{1+4x^6}}{2-x^3}\)
  3. \(\displaystyle \ds \lim_{x\to \infty} \cos x\)
  4. \(\displaystyle \ds \lim_{x\to \infty} \lrpar{\sqrt{9x^2 + x} - 3x}\)
  5. \(\displaystyle \ds \lim_{x\to \infty} \sqrt{x^2 + 2}\)
Solution.
  1. \(\displaystyle \ds \lim_{x\to -\infty} \dfrac{4x^3 + 6x^2 - 2}{2x^3 - 4x + 5} = 2\)
  2. \(\displaystyle \ds \lim_{x\to -\infty} \dfrac{\sqrt{1+4x^6}}{2-x^3}=2\)
  3. \(\ds \lim_{x\to \infty} \cos x\) does not exist, since \(\cos x\) oscillates between \(-1\) and \(1\) on its entire domain.
  4. \(\displaystyle \ds \lim_{x\to \infty} \lrpar{\sqrt{9x^2 + x} - 3x}=\dfrac{1}{6}\)
  5. \(\displaystyle \ds \lim_{x\to \infty} \sqrt{x^2 + 2} = \infty\)
Example 3.4.13.
A function \(f\) is a ratio of quadratic functions and has a vertical asymptote \(x=4\) and just one \(x-\)intercept, \(x=1\text{.}\) We know that \(f\) has a removable discontinuity at \(x=-1\text{,}\) and that \(\ds \lim_{x\to -1} f(x) = 2\text{.}\) Evaluate \(f(0)\) and find any horizontal asymptotes of \(f\text{.}\)
Solution.
\(f(0) = \dfrac{1}{2}\) and \(y=2\)
Example 3.4.14.
Sketch the function \(y = \dfrac{1 + 2x^2}{1+x^2}\) using the methods of Section 3.3 and this section.
Solution.
The graph of \(y = \dfrac{1 + 2x^2}{1+x^2}\) on the interval \([-2,2]\text{.}\)

Subsection 3.4.4 After Class Activities

Example 3.4.15.

Sketch the graph of a function that satisfies the conditions: \(f(1) = f'(1) = 0\text{,}\) \(\ds \lim_{x\to 2^+} f(x) = \infty\text{,}\) \(\ds \lim_{x\to 2^-} f(x) = -\infty\text{,}\) \(\ds \lim_{x\to 0} f(x) = -\infty\text{,}\) \(\ds \lim_{x\to -\infty} = \infty\text{,}\) \(\ds \lim_{x\to \infty} f(x) = 0\text{,}\) \(f''(x) \gt 0\) for \(x \gt 2\text{,}\) \(f''(x) \lt 0\) for \(x \lt 0\) and for \(0 \lt x \lt 2\text{.}\)
Solution.
Answers vary

Example 3.4.16.

Find \(\ds \lim_{x\to \infty} f(x)\text{,}\) if \(\dfrac{3x - 1}{x} \lt f(x) \lt \dfrac{3x^2 + 6}{x^2}\) for all \(x \gt 6\)
Solution.
3

Example 3.4.17.

A tank contains 5000 L of pure water. Brine containing 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Write an expression for the concentration of salt after \(t\) minutes (in grams per liter). What happens to the concentration as \(t\to\infty\text{?}\)
Solution.
\(C(t) = \dfrac{30t}{200+t}\text{.}\) As \(t\to\infty\text{,}\) \(C(t)\to 30\) g/L
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